A major drawback of FTN signaling is that the higher symbol rate induces inevitable and severe intersymbol interference (ISI) at the transmitter side. Consequently, a high complexity detector is usually required at the receiver side [1]}, [2]}, [3]}, [4]}. For example, the number of trellis states in a Bahl-Cocke-Jelinek-Raviv (BCJR) detector increases exponentially with the constellation size and number of ISI taps.
Moreover, for a coded FTN system, Turbo equalization is usually applied at the receiver, where iterations are performed between the BCJR detector and channel decoder. The overall detection/decoding complexity further increases with respect to the number of iterations.
Therefore, designing practical detectors with reduced-complexity is a major research topic for FTN signaling [5]}, [6]}, [7]}.
For example, two M-algorithm BCJR (M-BCJR) detectors were proposed for FTN signaling in [5]} based on the Ungerboeck observation model [9]}, and they show promising error performance for coded FTN systems by applying Turbo equalization.
Other than the BCJR algorithm, sum-product algorithms (SPA) have also been recognized as an efficient method to compute the marginal probabilities with a low complexity [10]} and it has been
widely used in the decoding of channel codes [11]} and signal processing [12]}. A soft-input soft-output (SISO) detection algorithm for linear ISI channels was proposed in [13]}, where the sum-product algorithm is applied to a factor graph (FG) of the coded FTN system.
The complexity of the algorithm is linear to the number of interferers during each iteration.
